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$If \tan^{- 1} \frac{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}} = α,$ then x² =

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a

cos 2α

b

sin 2α

c

tan 2α

d

cot 2α

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detailed solution

Correct option is B

Since, tan−1⁡1+x2−1−x21+x2+1−x2=α⇒1+x2−1−x21+x2+1−x2=tan⁡α1Using componendo and dividendo, we get21+x221−x2=1+tan⁡α1−tan⁡α=tan⁡π4+α⇒1+x21−x2=tan2⁡π4+α1Again using componendo and dividendo, we have1+x2−1−x21+x2+1−x2=tan2⁡(π/4+α)−1tan2⁡(π/4+α)+1⇒x2=−cos⁡π2+2α=sin⁡2α

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The value of $\cos \{\tan^{- 1} (\tan 2)\},$ is

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